In the previous article "Matrix of Linear Algebra", some basic concepts about matrices have been introduced. This article mainly makes a further summary on solving the inverse matrix.
Minor
Let's first look at an example to intuitively understand what is a sub-form (Minor, the English Minor will be used later, and the Chinese translation is messy).
minor example
In this example (we assume the matrix is A), we see that the minor of A[1,1] is the determinant of the matrix left after deleting the row and column where A[1,1] is located. Suppose we put A [ 1 ,1] The minor is recorded as M[1,1], in this example it is
M1,1
In the same way, the minor of A[i, j] is the determinant of the remaining matrix after removing the i-th row and j-th column.
Matrix of Minors
We now know how to solve the minor of an element. Now we solve the minors of all the elements of a matrix and get a new matrix called matrix of minors. As shown in the figure below, it is the minor matrix of matrix A in our example
minors of A
Matrix of Cofactors
First of all, we will introduce Cofactor. We record the cofactor of M[i,j] as C[i,j]. We can have the following formula:
cofactor
Through this calculation formula, we can get all the C corresponding to M, which also forms a matrix, which is the matrix of cofactors. Let's take our example above to see how to get the matrix of cofactors, denoted as C
matrix of cofactors
When we have the matrix of cofactors, we can calculate the determinant of A |A|. The calculation process is to multiply the value A[1,j] of the first row of A by the corresponding cofactorC[1,j] , and then add the results
|A| = 1x(-3) + 2x6 + 3x(-3)=0
When |A|=0, we call A a singular matrix, and if |A|!=0, we call A a non-singular matrix. A singular matrix has no inverse. The last thing I want to say is that I originally wanted to find the inverse matrix, but unfortunately found a singular matrix, please forgive me :(
Adjoint MatrixAdjugate Matrix
The adjoint matrix is the matrix obtained after transposing the matrix of cofactors. We call it the adjoint matrix of A and denote it as adj(A). The so-called transposition is to exchange the value of [i, j] with the value of [j, i]. The specific example is as follows:
adjugate matrix
Note: This example is not very obvious, in fact all the values of C[i,j] and C[j,i] are exchanged, such as C[2,3] and C[3,2]
Since the example A in this article is a singular matrix, there is no inverse matrix, but if it is a non-singular matrix, we can obtain the inverse matrix according to the previous formula.
Inverse matrix calculation
elementary transformation
In addition to the above method to solve the inverse matrix, you can also use a more intuitive method to solve it. This is the elementary transformation. The principle is based on the principle that the inverse of A multiplied by A is equal to the identity matrix I. Interested students can see it in the reference link. video.
reference:
1, Khan Open Class
2, minor introduction in wikipedia
3, Wyman's technical blog